where the sum is over all the disjoint loops \({\mathcal {L}}\) 's in \(\Gamma \) .
The pole of the SGF thus corresponds to the zeros of the 2D Ising amplitude for the same graph.
The dependence of the scaleless spin network states on the spinors can also be rewritten as the dependence on the link couplings (defined below in REF ) as one-to-one correspondence to the edge couplings in the Ising mode.
Thus this duality directly gives an exact and closed expression of the scaleless spin network for a general three-valent graph \(\Gamma \)  (see [1]}). The exact expression is also proven directly in [2]}, [3]}.
A similar result was also found for a square 2D lattice [4]}
